Design of small RF antennas

Where do the geometries come from? I've been looking at embedded PCB antennas and also modules that attach to PCBs and they always have exotic looking notches and curves. Some are 3D while others are flat 2D styles. How does one actually arrive at such geometries mathematically? They seem so random and I know that their are exact subtleties to why the specific combination of geometries are in place, and even then, these geometries depend on the materials being used.

In the cellular industry, is it mostly intuitive knowledge combined with an iterative design approach that these designers come to their final antenna design, or do they use special design tools/simulations or mathematics/physics theory to get there? I know its probably a combination, but I'm wondering what it takes to actually design an antenna.

I'm particularly interested in the design of cellular quad band passive antennas, so if anyone has any experience or suggestions on where to start, I would appreciate it greatly.

For broadband applications they are usualy fractals. There are ltos of ways to generate fractals.
I think most applications are designed by picking some fractal parameters and then making a real or computer simulated antenna, testing it and then adjusting the paramters.
I don't know you can calculate what fractal is best from first principles.

The principles describing the behavior of electromagnetic waves are the Maxwell's equations; that's where we start. More complicated solutions can be approximated by sophisticated software simulations.

A quad band cellular antenna for instance is designed to be sensitive to bands of interest, like 1.8 and 1.9 GHz, and some bands in 800 MHz range. The design could have two antennas duplexed together where one is sensitive to the higher frequencies, and one to lower frequencies. There is no need to pick up anything in between.

You enter dimensions for wires in a coordinate form that lets you set the spacing between wires as well. Wires can be any diameter so they can be pipes or solid rods or actually thin wires.
A wire is regarded as a number of smaller chunks called segments and you specify how many of these there are in a wire. You get greater accuracy of predicted effects if you have more segments.

This demo limits you to 20 segments but the commercial version allows hundreds of segments.
This is ample for simple antennas.
Sample antenna designs are included.

Other programs ask for a frequency and then just present you with the dimensions for building the antenna.
Building the antenna nearly always involves a bit of adjustment of the computer generated dimensions, but they are a good guide as a start.